Recently, Bonnefoy et al. (J. Fluid Mech., vol. 805, 2016, R3) studied the resonant interaction of oblique surface gravity waves in a large $50~\text{m}\times 30~\text{m}\times 5~\text{m}$ wave basin. Their experimental results are in excellent quantitative agreement with predictions of the weakly nonlinear wave theory, and provide additional evidence to the strength of this widely used mathematical formulation. In this article, the reader is introduced to the many facets of the weakly nonlinear theory for surface gravity waves, and to its current and possible future applications, deterministic as well as stochastic.

Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.

This paper reports on an experimental study of the influence of the Strouhal number (0.011, 0.022 and 0.044) and orifice-to-plate distances (2, 4 and 6 orifice diameters) on the flow field of an impinging zero-net-mass-flux jet at a Reynolds number equal to 35 000. These jets are generated by a reciprocating piston that oscillates in a cavity behind a circular orifice. Instantaneous two-dimensional in-plane velocity fields are measured in a plane containing the orifice axis using multigrid/multipass cross-correlation digital particle image velocimetry. These measurements have been used to investigate the mean flow quantities and turbulent statistics of the impinging zero-net-mass-flux jets. In addition, the vortex ring behaviour is analysed via its trajectory and azimuthal vorticity as well as the saddle point excursion, the flow rate and entrainment. The behaviour of all these quantities depends on the Strouhal number and the orifice-to-plate distance because the former governs the presence and the relative importance of the vortex ring and the trailing jet on the flow field and the latter delimits the downstream evolution of these structures.

A reactive control technique with localised actuators and sensors is used to delay the transition to turbulence in a flat-plate boundary-layer flow. Through extensive direct numerical simulations, it is shown that an adaptive technique, which computes the control law on-line, is able to significantly reduce skin-friction drag in the presence of random three-dimensional perturbation fields with linear and weakly nonlinear behaviour. An energy budget analysis is performed in order to assess the net energy saving capabilities of the linear control approach. When considering a model of the dielectric-barrier-discharge (DBD) plasma actuator, the energy spent to create appropriate actuation force inside the boundary layer is of the same order as the energy gained from reducing skin-friction drag. With a model of an ideal actuator a net energy gain of three orders of magnitude can be achieved by efficiently damping small-amplitude disturbances upstream. The energy analysis in this study thus provides an upper limit for what we can expect in terms of drag-reduction efficiency for linear control of transition as a means for drag reduction.

The near-field breakup and interaction of a hollow-cone liquid sheet with coannular swirling air flow have been examined using high-speed diagnostics. Time-resolved PIV (particle image velocimetry; $3500~\text{frames}~\text{s}^{-1}$) is employed to capture the spatio-temporal behaviour of the swirling air flow field. The combined liquid–gas phase interaction is visualized with the help of high-speed ($20\,000~\text{frames}~\text{s}^{-1}$) shadowgraphy. In this study, the transition from weak to strong spray–swirl interaction is explained based on the momentum ratio. Proper orthogonal decomposition (POD) is implemented on instantaneous PIV and shadowgraphy images to extract the energetic spatial eigenmodes and characteristic modal frequencies. The POD results suggest the dominance of the KH (Kelvin–Helmholtz) instability mechanism (pure axial shear, axial plus azimuthal shear) in swirl–spray interaction. In addition, linear stability analysis also shows the destabilization of the liquid–air interface caused by KH waves ($\unicode[STIX]{x1D706}_{p}$), which arises from the formation of a vorticity layer of thickness $\unicode[STIX]{x1D6FF}_{g}$ near the liquid–air interface. The frequency values obtained from the primary KH wavelength ($\unicode[STIX]{x1D706}_{p}$) exhibit good agreement with the POD modal frequencies. Scaling laws are proposed to elucidate the relationships between the global length scales (breakup length, spray spread) and the primary wavelength. The breakup length scale and liquid sheet oscillations are meticulously analysed in the time domain to reveal the breakup dynamics of the liquid sheet. Furthermore, the large-scale coherent structures of the swirl flow exhibit different sheet breakup phenomena in the spatial domain. For instance, flapping breakup is induced by the central toroidal recalculation zone in the swirling flow field. Finally, the ligament formation mechanism and its diameter, i.e. the size of first-generation droplets, are measured with phase Doppler interferometry. The measured sizes scale reasonably with KH waves.

A model is developed for the extrusion in the direction of gravity of a slender fluid cylinder from a die of arbitrary shape. Both gravity and surface tension act to stretch and deform the geometry. The model allows for an arbitrary but prescribed viscosity profile, while the effects of extrudate swell are neglected. The solution is found efficiently through the use of a carefully selected axial Lagrangian coordinate and a transformation to a reduced-time variable. Comparisons between the model and extruded glass microstructured optical fibre preforms show that surface tension has a significant effect on the geometry but the model does not capture all of the behaviour observed in practice. Experimental observations are used in conjunction with the model to argue that some deformation, due neither to surface tension nor gravity, occurs in or near the die exit. Methods are considered to overcome deformation due to surface tension.

Dynamic shape change of the octopus mantle during fast jet escape manoeuvres results in added mass energy recovery to the energetic advantage of the octopus, giving escape thrust and speed additional to that due to jetting alone. We show through numerical simulations and experimental validation of overall wake behaviour, that the success of the energy recovery is highly dependent on shrinking speed and Reynolds number, with secondary dependence on shape considerations and shrinking amplitude. The added mass energy recovery ratio $\unicode[STIX]{x1D702}_{ma}$, which measures momentum recovery in relation to the maximum momentum recovery possible in an ideal flow, increases with increasing the non-dimensional shrinking parameter $\unicode[STIX]{x1D70E}^{\ast }={\dot{a}}_{max}/U\sqrt{\mathit{Re }_{0}}$, where ${\dot{a}}_{max}$ is the maximum shrinking speed, $U$ is the characteristic flow velocity and $\sqrt{\mathit{Re }_{0}}$ is the Reynolds number at the beginning of the shrinking motion. An estimated threshold $\unicode[STIX]{x1D70E}^{\ast }\approx 10$ determines whether or not enough energy is recovered to the body to produce net thrust. Since there is a region of high transition for $10<\unicode[STIX]{x1D70E}^{\ast }<30$ where the recovery performance varies widely and for $\unicode[STIX]{x1D70E}^{\ast }>100$ added mass energy is recovered at diminishing returns, we propose a design criterion for shrinking bodies to be in the range of $50<\unicode[STIX]{x1D70E}^{\ast }<100$, resulting in 61–82 % energy recovery.

The nature of the flow below the cloud level on Jupiter and Saturn is still unknown. Relating the flow on these planets to perturbations in their density field is key to the analysis of the gravity measurements expected from both the Juno (Jupiter) and Cassini (Saturn) spacecrafts during 2016–2018. Both missions will provide latitude-dependent gravity fields, which in principle could be inverted to calculate the vertical structure of the observed cloud-level zonal flow on these planets. Theories to date connecting the gravity field and the flow structure have been limited to potential theories under a barotropic assumption, or estimates based on thermal wind balance that allow baroclinic wind structures to be analysed, but have made simplifying assumptions that neglected several physical effects. These include the effects of the deviations from spherical symmetry, the centrifugal force due to density perturbations and self-gravitational effects of the density perturbations. Recent studies attempted to include some of these neglected terms, but lacked an overall approach that is able to include all effects in a self-consistent manner. The present study introduces such a self-consistent perturbation approach to the thermal wind balance that incorporates all physical effects, and applies it to several example wind structures, both barotropic and baroclinic. The contribution of each term is analysed, and the results are compared in the barotropic limit with those of potential theory. It is found that the dominant balance involves the original simplified thermal wind approach. This balance produces a good order-of-magnitude estimate of the gravitational moments, and is able, therefore, to address the order one question of how deep the flows are given measurements of gravitational moments. The additional terms are significantly smaller yet can affect the gravitational moments to some degree. However, none of these terms is dominant so any approximation attempting to improve over the simplified thermal wind approach needs to include all other terms.

The Reynolds-number dependence of turbulent channel flow over two irregular rough surfaces, based on scans of a graphite and a grit-blasted surface, is studied by direct numerical simulation. The aim is to characterise the changes in the flow in the immediate vicinity of and within the rough surfaces, an area of the flow where it is difficult to obtain experimental measurements. The average roughness heights and spatial correlation of the roughness features of the two surfaces are similar, but the two surfaces have a significant difference in the skewness of their height distributions, with the graphite sample being positively skewed (peak-dominated) and the grit-blasted surface being negatively skewed (valley-dominated). For both cases, numerical simulations were conducted at seven different Reynolds numbers, ranging from $Re_{\unicode[STIX]{x1D70F}}=90$ to $Re_{\unicode[STIX]{x1D70F}}=720$. The positively skewed surface gives rise to higher friction factors than the negatively skewed surface in all cases. For the highest Reynolds numbers, the flow has values of the roughness function $\unicode[STIX]{x0394}U^{+}$ well in excess of $7$ for both surfaces and the bulk flow profile has attained a constant shape across the full height of the channel except for the immediate vicinity of the roughness, which would indicate fully rough flow. However, the mean flow profile within and directly above the rough surface still shows considerable Reynolds-number dependence and the ratio of form to viscous drag continues to increase, which indicates that at least for some types of rough surfaces the flow retains aspects of the transitionally rough regime to values of $\unicode[STIX]{x0394}U^{+}$ or $k^{+}$ well in excess of the values conventionally assumed for the transitionally to fully rough threshold. This is also reflected in the changes that the near-wall flow undergoes as the Reynolds number increases: the viscous sublayer, within which the surface roughness is initially buried, breaks down and regions of reverse flow intensify. At the highest Reynolds numbers, a layer of near-wall flow is observed to follow the contours of the local surface. The distribution of thickness of this ‘blanketing’ layer has a mixed scaling, showing that viscous effects are still significant in the near-wall flow.

The deformation of a viscous liquid droplet suspended in another liquid and subject to an applied electric field is a classic multiphase flow problem best described by the Melcher–Taylor leaky dielectric model. The main assumption of the model is that any net charge in the system is concentrated on the interface between the two liquids as a result of the jump in Ohmic currents from the bulk. Upon application of the field, the drop can either attain a steady prolate or oblate shape with toroidal circulating flows both inside and outside arising from tangential stresses on the interface due to action of the field on the surface charge distribution. Since the pioneering work of Taylor (Proc. R. Soc. Lond. A, vol. 291, 1966, pp. 159–166), there have been numerous computational and theoretical studies to predict the deformations measured in experiments. Most existing theoretical models, however, have either neglected transient charge relaxation or nonlinear charge convection by the interfacial flow. In this work, we develop a novel small-deformation theory accurate to second order in electric capillary number $O(Ca_{E}^{2})$ for the complete Melcher–Taylor model that includes transient charge relaxation, charge convection by the flow, as well as transient shape deformation. The main result of the paper is the derivation of coupled evolution equations for the induced electric multipoles and for the shape functions describing the deformations on the basis of spherical harmonics. Our results, which are consistent with previous models in the appropriate limits, show excellent agreement with fully nonlinear numerical simulations based on an axisymmetric boundary element formulation and with existing experimental data in the small-deformation regime.

Motivated by systems in which droplets grow and shrink in a turbulence-driven supersaturation field, we investigate the problem of turbulent condensation in a general manner. Using direct numerical simulations, we show that the turbulent fluctuations of the supersaturation field offer different conditions for the growth of droplets which evolve in time due to turbulent transport and mixing. Based on this, we propose a Lagrangian stochastic model for condensation and evaporation of small droplets in turbulent flows. It consists of a set of stochastic integro-differential equations for the joint evolution of the squared radius and the supersaturation along the droplet trajectories. The model has two parameters fixed by the total amount of water and the thermodynamic properties, as well as the Lagrangian integral time scale of the turbulent supersaturation. The model reproduces very well the droplet size distributions obtained from direct numerical simulations and their time evolution. A noticeable result is that, after a stage where the squared radius simply diffuses, the system converges exponentially fast to a statistical steady state independent of the initial conditions. The main mechanism involved in this convergence is a loss of memory induced by a significant number of droplets undergoing a complete evaporation before growing again. The statistical steady state is characterized by an exponential tail in the droplet mass distribution. These results reconcile those of earlier numerical studies, once these various regimes are considered.

A liquid jet falling by gravity ultimately destabilizes by capillary forces. Accelerating as it falls, the jet thins and stretches, causing a capillary instability to develop on a spatially varying substrate. We discuss quantitatively the interplay between instability growth, jet thinning and longitudinal stretching for two kinds of perturbations, either solely introduced at the jet nozzle exit or affecting the jet all along its length. The analysis is conducted for any values of the liquid properties for a sufficiently large flow rate. In all cases, we determine the net gain of the most dangerous perturbation for all downstream distances, thus predicting the jet length, the wavelength at breakup and the resulting droplet size.

In this paper we revisit the Landau and Levich analysis of a coating flow in the case where the flow in the thin liquid film is supported by a Rayleigh surface acoustic wave (SAW), propagating in the solid substrate. Our theoretical analysis reveals that the geometry of the film evolves under the action of the propagating SAW in a manner that is similar to the evolution of films that are being deposited using the dip coating technique. We show that in a steady state the thin-film evolution equation reduces to a generalized Landau–Levich equation with the dragging velocity, imposed by the SAW, depending on the local film thickness. We demonstrate that the generalized Landau–Levich equation has a branch of stable steady state solutions and a branch of unstable solutions. The branches meet at a saddle-node bifurcation point corresponding to the threshold value of the SAW intensity. Below the threshold value no steady states were found and our numerical computations suggest a gradual thinning of the liquid film from its initial geometry.

It is known that large-scale streamwise velocity-fluctuating structures ($u^{\prime }$) are frequently observed in the log region of a zero pressure gradient turbulent boundary layer, and that these motions significantly influence near-wall small-scale $u^{\prime }$-structures by modulating the amplitude (Hutchins & Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28; Mathis et al., J. Fluid Mech., vol. 628, 2009, pp. 311–337). In the present study, we provide evidence that the spatial organization of large-scale structures in the log region is significantly influenced by the strength of adverse pressure gradients in turbulent boundary layers based on a direct numerical simulation dataset. For a mild adverse pressure gradient boundary layer flow, groups of hairpin vortices are coherently aligned in the streamwise direction to form hairpin vortex packets, and streamwise merging events of the induced large-scale $u^{\prime }$-structures create a larger streamwise length scale of structures than that for a zero pressure gradient boundary layer flow. As the pressure gradient strength increases further, however, the formation of hairpin packets is continuously suppressed, and large-scale motions are consequently not concatenated to create a longer motion, resulting in a significant reduction of the streamwise coherence of large-scale structures in the log layer. Although energy spectrum maps for $u^{\prime }$-structures show that the large-scale energy is continuously intensified above the log layer with an increase in the pressure gradient, amplitude modulation of the near-wall small-scale motions is dominantly induced by log region large-scale structures for adverse pressure gradient flows. Conditional averaged flow fields with large-scale Q2 and Q4 events indicate that large-scale counter-rotating roll modes play an important role in organizing the flows under the pressure gradients, and the large-scale roll modes associated with Q4 events are more enhanced in the outer layer than those associated with Q2 events, reducing the streamwise coherence of the vortices in a packet.

We consider the time dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,\unicode[STIX]{x1D714}}$ and $D_{m,\unicode[STIX]{x1D703}}$ of the vorticity $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ and the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$, where $\unicode[STIX]{x1D703}=\log (\unicode[STIX]{x1D70C}^{\ast }/\unicode[STIX]{x1D70C}_{0}^{\ast })$, in a buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. Fluid Mech., vol. 591, 2007, pp. 43–71). Here, $\unicode[STIX]{x1D70C}^{\ast }(\boldsymbol{x},t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $\unicode[STIX]{x1D70C}_{2}^{\ast }>\unicode[STIX]{x1D70C}_{1}^{\ast }$, and $\unicode[STIX]{x1D70C}_{0}^{\ast }$ is a reference normalization density. Using data from the publicly available Johns Hopkins turbulence database, we present evidence that the $L^{2}$-spatial average of the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ can reach extremely large values at intermediate times, even in flows with low Atwood number $At=(\unicode[STIX]{x1D70C}_{2}^{\ast }-\unicode[STIX]{x1D70C}_{1}^{\ast })/(\unicode[STIX]{x1D70C}_{2}^{\ast }+\unicode[STIX]{x1D70C}_{1}^{\ast })=0.05$, implying that very strong mixing of the density field at small scales can arise in buoyancy-driven turbulence. This large growth raises the possibility that the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ might blow up in a finite time.

The ability of acoustic microstreaming to cause a pair of particles to attract or repel is investigated. Expanding the flow around two spheres in terms of a small-amplitude parameter measuring the amplitude of the forcing, the leading order is an oscillating flow field with zero mean representing the effect of the applied acoustic field, while the second-order correction contains a steady streaming component. A modal decomposition in the azimuthal direction reduces the problem to a few linear problems in a two-dimensional domain corresponding to the meridional ($r,z$) plane. The analysis computes both the intricate flow fields and the mean forces felt by both spheres. If the spheres are aligned obliquely with respect to the oscillating flow, they experience a lateral force which realigns them into a transverse configuration. In this transverse configuration, they experience an axial force which can be either attractive or repulsive. At high frequencies the force is always attractive. At low frequencies, it is repulsive. At intermediate frequencies, the force is attractive at large distances and repulsive at small distances, leading to the existence of a stable equilibrium configuration.

We investigate the dynamic stability of stratified flow configurations characteristic of hydraulically controlled downslope flow over topography. Extraction of the correct ‘base state’ for stability analysis from spatially and temporally evolving flows that exhibit instability is not easy since the observed flow in most cases has already been modified by nonlinear interactions between the instability modes and the mean flow. Analytical studies, however, can yield steady solutions under idealized conditions which can then be analysed for stability. Following the latter approach, we study flow profiles whose essential character is determined by recently obtained solutions of Winters & Armi (J. Fluid Mech., vol. 753, 2014, pp. 80–103) for topographically controlled stratified flows. Their condition of optimal control necessitates a streamline bifurcation which then naturally produces a stagnant isolating layer overlying an accelerating stratified jet in the lee of the topography. We show that the inclusion of the isolating layer is an essential component of the stability analysis and further clarify the nature and mechanism of the instability in light of the wave-interaction theory. The spatial stability problem is also briefly examined in order to estimate the downstream location where finite-amplitude features might be manifested in streamwise slowly varying flows over topography.

We examine the effectiveness of the generalised quasilinear (GQL) approximation introduced by Marston et al. (Phys. Rev. Lett., vol. 116 (21), 2016, 214501). This approximation splits the variables into large and small scales in directions where there is a translational symmetry and removes nonlinear interactions involving only small scales. We utilise as a paradigm problem three-dimensional, turbulent, rotating Couette flow. We compare the results obtained from direct numerical solution of the equations with those from quasilinear (QL) and GQL calculations. In this three-dimensional setting, there is a choice of cutoff wavenumber for the GQL approximation both in the streamwise and in the spanwise directions. We demonstrate that the GQL approximation significantly improves the accuracy of mean flows, spectra and two-point correlation functions over models that are quasilinear in any of the translationally invariant directions, even if only a few streamwise and spanwise modes are included. We argue that this provides significant support for a programme of direct statistical simulation utilising the GQL approximation.

We investigate the effect of an external freshwater plume on the dissolution of a vertical ice wall in salty water using laboratory experiments. We measure the plume velocity, the ablation velocity of the ice and the temperature at the ice wall. The freshwater volume flux, $Q_{s}$, is varied between experiments to determine where the resultant wall plume transitions from being dominated by the distributed buoyancy flux due to dissolution of the ice, to being dominated by the initial buoyancy flux, $B_{s}$. We find that when $B_{s}$ is significantly larger than the distributed buoyancy flux from dissolution, the plume velocity is uniform with height and is proportional to $B_{s}^{1/3}$, the interface temperature is independent of $B_{s}$, and the ablation velocity increases with $B_{s}$.

The unsteady response of a water free surface to a localized pressure source moving at constant speed $U$ in the range $0.95c_{min}\lesssim U\leqslant 1.02c_{min}$, where $c_{min}$ is the minimum phase speed of linear gravity–capillary waves in deep water, is investigated through experiments and numerical simulations. This unsteady response state, which consists of a V-shaped pattern behind the source, and features periodic shedding of pairs of depressions from the tips of the V, was first observed qualitatively by Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) and called state III. In the present investigation, cinematic shadowgraph and refraction-based techniques are utilized to measure the temporal evolution of the free-surface deformation pattern downstream of the source as it moves along a towing tank, while numerical simulations of the model equation described by Cho et al. (J. Fluid Mech., vol. 672, 2011, pp. 288–306) are used to extend the experimental results over longer times than are possible in the experiments. From the experiments, it is found that the speed–amplitude characteristics and the shape of the depressions are nearly the same as those of the freely propagating gravity–capillary lumps of inviscid potential theory. The decay rate of the depressions is measured from their height–time characteristics, which are well fitted by an exponential decay law with an order one decay constant. It is found that the shedding period of the depression pairs decreases with increasing source strength and speed. As the source speed approaches $c_{min}$, this period tends to approximately 1 s for all source magnitudes. At the low-speed boundary of state III, a new response with unsteady asymmetric shedding of depressions is found. This response is also predicted by the model equation.